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Aleksy Wróblewski 2018-06-27 17:24:40 +00:00
parent 128cac8f55
commit 3ff8beeed6
2 changed files with 199 additions and 0 deletions

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from poly import Polynomial
from sys import argv
from ast import literal_eval
from fractions import gcd
class QuotientRing():
def __init__(self, f, m):
self.f = Polynomial(f, m)
self.m = m
self.remainders = self.remainders()
self.reversibles = self.reversibles()
self.zero_divisors = self.zero_divisors()
self.idempotent = self.idempotent()
self.nilpotent = self.nilpotent()
def remainders(self): #n - exponent
rems = [] #lista reszt
m = self.m
t = [0]
i = 0
while len(t) < len(self.f.poly):
rems.append(Polynomial(t, m))
i = (i + 1) % m
t[0] = i
if i == 0:
if len(t) == 1:
t.append(1)
else:
t[1] += 1
for j in range(1, len(t)):
if t[j] == 0 or t[j] % m != 0:
break
temp = t[j] % m
t[j] = 0
if temp == 0:
if (j + 1) < len(t):
t[j+1] += 1
else:
t.append(1)
return rems
def reversibles(self):
return [ rem for rem in self.remainders if len(rem.poly_gcd(self.f).poly) == 1 ]
#dopelnienie elementow odwracalnych
def zero_divisors(self):
return [ rem for rem in self.remainders if rem not in self.reversibles ]
def idempotent(self):
idems = []
for rem in self.remainders:
if (rem * rem / self.f) == (rem / self.f):
idems.append(rem)
try:
if idems[0].poly == []: #implementacja wielomianow ucina zera
idems[0].poly = [0]
except IndexError:
return idems
return idems
def nilpotent(self):
nils = []
phi = len([ i for i in range(1, self.m) if gcd(i, self.m) == 1 ])
for zero_div in self.zero_divisors:
for i in range(self.m):
if len((zero_div ** i / self.f).poly) == 0:
nils.append(zero_div)
break
return nils
def main():
m = int(argv[1])
f = literal_eval(argv[2])
qr = QuotientRing(f, m)
out = [
[ rev.poly for rev in qr.reversibles ],
[ zero_div.poly for zero_div in qr.zero_divisors ],
[ nil.poly for nil in qr.nilpotent ],
[ idem.poly for idem in qr.idempotent ]
]
print(*out, sep='\n')
if __name__ == '__main__':
main()

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from fractions import gcd
class Polynomial():
def __init__(self, lst, mod):
self.poly = list(map(lambda x: x % mod, lst))
self.mod = mod
self.normalize()
def normalize(self):
while self.poly and self.poly[-1] == 0:
self.poly.pop()
#zwraca jednomian stopnia n
@staticmethod
def Monomial(n, c, mod):
zeros = [0]*n
zeros.append(c)
return Polynomial(zeros, mod)
def __add__(self, p2):
p1 = self
len_p1, len_p2= len(p1.poly), len(p2.poly)
res = [0] * max(len_p1, len_p2)
if len_p1 > len_p2:
for _ in range(len_p1-len_p2):
p2.poly.append(0)
else:
for _ in range(len_p2-len_p1):
p1.poly.append(0)
for i in range(len(res)):
res[i] = (p1.poly[i] + p2.poly[i]) % self.mod
return Polynomial(res, self.mod)
def __sub__(self, p2):
p1 = self
res = []
len_p2 = len(p2.poly)
for i in range(len(p1.poly)):
if i < len_p2:
res.append(p1.poly[i] - p2.poly[i] % self.mod)
else:
res.append(p1.poly[i])
return Polynomial(res, self.mod)
def __mul__(self, p2):
res = [0]*(len(self.poly)+len(p2.poly)-1)
for i, x1 in enumerate(self.poly):
for j, x2 in enumerate(p2.poly):
res[i+j] += x1 * x2 % self.mod
return Polynomial(res, self.mod)
def __eq__(self, p2):
p1 = self
return p1.poly == p2.poly and p1.mod == p2.mod
def __pow__(self, n):
p1 = self
for i in range(n):
p1 = p1 * p1
return p1
def __truediv__(self, p2):
p1 = self
m = self.mod
if len(p1.poly) < len(p2.poly):
return p1
if len(p2.poly) == 0:
raise ZeroDivisionError
divisor_coeff = p2.poly[-1]
divisor_exp = len(p2.poly) - 1
while len(p1.poly) >= len(p2.poly):
max_coeff_p1 = p1.poly[-1] #wspolczynnik przy najwyzszej potedze
try:
tmp_coeff = modDiv(max_coeff_p1, divisor_coeff, m)
except ZeroDivisionError as e:
raise e
tmp_exp = len(p1.poly)-1 - divisor_exp
tmp = [0] * tmp_exp
tmp.append(tmp_coeff)
sub = Polynomial(tmp, m) * p2
p1 = p1 - sub
p1.normalize()
return Polynomial(p1.poly, m)
def poly_gcd(self, p2):
p1 = self
try:
divisible = p2
except ZeroDivisionError as e:
raise e
if p2.poly == []:
return p1
return p2.poly_gcd(p1 / p2)
def modDiv(a, b, m): # a*b^-1 (mod m)
if gcd(b, m) != 1:
raise ZeroDivisionError
else:
return (a * modinv(b, m)) % m
#rozszerzony algorytm euklidesa
def egcd(a, b):
if a == 0:
return (b, 0, 1)
else:
g, y, x = egcd(b % a, a)
return (g, x - (b // a) * y, y)
def modinv(a, m):
g, x, y = egcd(a, m)
return x % m